Reduced-Order Model for Cell Volume Homeostasis: Application to Aqueous Humor Production

Abstract

The ability of a cell to keep its volume constant irrespective of intra- and extracellular conditions is essential for cellular homeostasis and survival. The purpose of this study is to elaborate a theoretical model of cell volume homeostasis and to apply it to a simulation of human aqueous humor (AH) production. The model assumes a cell with a spherical shape and only radial deformation satisfying the property that the cell volume in rest conditions equals that of the cell couplets constituting the ciliary epithelium of the human eye. The cytoplasm is described as a homogeneous mixture containing fluid, ions, and neutral solutes whose evolution is determined by net production mechanisms occurring in the intracellular volume and by water and solute exchange across the membrane. Averaging the balance equations over the cell volume leads to a coupled system of nonlinear ordinary differential equations (ODEs) which are solved using the _θ -method and the Matlab function ode15s. Simulation tests are conducted to characterize the set of parameters corresponding to baseline conditions in AH production. The model is subsequently used to investigate the relative importance of (a) impermeant charged proteins; (b) sodium-potassium (Na+/K+) pumps; (c) carbonic anhydrase (CA) in the AH production process; and (d) intraocular pressure. Results suggest that (a) and (b) play a role; (c) lacks significant weight, at least for low carbon dioxide values; and (d) plays a role for the elevated values of intraocular pressure. Model results describe a higher impact from charged proteins and Na+/K+ ATPase than CA on AH production and cellular volume. The computational virtual laboratory provides a method to further test in vivo experiments and machine learning-based data analysis toward the prevention and cure of ocular diseases such as glaucoma.

Keywords: aqueous humor production; cell volume; homeostasis; homogeneous mixtures; mathematical modeling.

Figure A1.

Red solid line: a plot of $⟨c_{\alpha }⟩$ for $\tau =[0:20]$ and $X_{\alpha }(\tau )=\tau -10$, with $\tau$ being a dimensionless time. The endpoint values of $c_{\alpha }$ are $c_{\alpha ,in}\left(\tau \right)=1mM$ and $c_{\alpha ,ex}\left(t\right)=5\mathrm{m}\mathrm{M}$ for every $\tau \in [0:20]$. Black dashed line: the arithmetic average of $c_{\alpha }$.

Figure A2.

Schematic representation of intracellular reactions and transmembrane transport mechanisms. MIT: mitochondrium. ATP: adenosinetriphosphate. CA: carbonic anhydrase. Exchangers perform multiple ion transport across membrane.

Figure 1.

A schematic representation of the processes involved in AH dynamics. (1): AH production; (2): AH flow; (3): AH outflow. Reprinted from R. Ramakrishnan et al, Diagnosis & Management of Glaucoma, Chapter 9 Aqueous Humor Dynamics, 10.5005/jp/books/11801_9, (2013) [21]; used in accordance with the Creative Commons Attribution (CC BY) license.

Figure 2.

Left panel: a light micrograph of the ciliary body epithelium. It consists of two epithelial cell layers: a non-pigmented inner layer and an outer pigmented layer. Under the epithelium, there is a highly vascularized stroma. Reprinted from [23]; used in accordance with the Creative Commons Attribution (CC BY) license. Right panel: a compartmental representation of the CE cell couplet.

Figure 3.

The “equivalent cell”. Solid line: initial cell configuration. Dashed line: deformed cell configuration. The cyan arrows indicate water flow. The cell is increasing its volume (cell swelling).

Figure 4.

A schematic representation of the structure of the cell membrane. AQP: aquaporin (cyan). The ion channel is drawn in green. Water molecules (red and dark blue), charged solutes (magenta), and neutral solutes (brown) are illustrated. The lipid constituent is drawn in yellow. The AQP is selective to water molecules whereas the ion channel permits the co-transport of ions and water.

Figure 5.

A three-dimensional schematic representation of an aquaporin. The cylindrical domain ${\omega }_p$ is the pore channel, $t_m$ is the membrane thickness, and $r_p$ is the aquaporin radius.

Figure 6.

Transmembrane electric potential.

Figure 7.

Left panel: a plot of $f(x)$. Right panel: a plot of $𝒱(t)/{𝒱}_{\mathit{ref}}$ in the time interval [0, 10] s. The values of the input data are as follows: $R_{\mathit{cell}}=10\times {10}^{-6}\mathrm{m}$, ${\mathrm{\Phi }}^{AQP}=0.5$ ; $\bar{v}=-30\times {10}^{-6}\mathrm{m}{\mathrm{s}}^{-1}$ ; ${\kappa }_a=1{\mathrm{s}}^{-1}$ ; and $k_d=3{\mathrm{s}}^{-1}$.

Figure 8.

Left panel: a plot of $𝒱(t)/{𝒱}_{\mathit{ref}}$ in the time interval [0, 1] s. The value of fluid velocity (expressed in μms⁻¹) is indicated for each computed normalized cell volume. Right panel: a plot of $𝒱(t)/{𝒱}_{\mathit{ref}}$ in the time interval [0, 1] s.

Figure 9.

Left panel: a plot of the water volume net production rate ${ℛ}_w(t)$ in the time interval [0, 10] s. Right panel: a plot of $𝒱(t)/{𝒱}_{\mathit{ref}}$ in the time interval [0, 1] s in the case where $ℛ(t)=0$ for every $t\in \left[0,1\right]\mathrm{s}$. Fluid velocity varies in the range [−30 + 30] × 10⁻⁶ ms⁻¹. The arrow indicates velocity increase from negative to positive values.

Figure 10.

Left panel: blue curve, $c_{{\mathrm{H}}^{+},in}(t)$ ; red curve, $\mathrm{p}{\mathrm{H}}_{,\mathit{in}}\left(t\right),t\in \left[0,50\cdot {10}^{-12}\right]\mathrm{s}$. Right panel: blue curve, $v_{\mathit{cell},n}(t)$ ; red curve, $\mathrm{\Delta }{𝒱}_{\%}\left(t\right)$, $t\in \left[0,50\cdot {10}^{-12}\right]\mathrm{s}$. Middle panel (bottom): total AH volumetric flow rate $Q_{\mathrm{A}\mathrm{H}}(t)$ for $t\in \left[0,50\cdot {10}^{-12}\right]\mathrm{s}$.

Figure 11.

Top left panel: blue curve, $c_{{\mathrm{C}\mathrm{O}}_{2,in}}(t)$ ; red curve, ${\mathrm{H}}_2{\mathrm{C}\mathrm{O}}_{3,in}\left(t\right)$, $t\in \left[0,5\right]\mathrm{s}$. Top right panel: blue curve, ${\mathrm{H}}_{\mathit{in}}^{+}(t)$ ; red curve, ${\mathrm{p}\mathrm{H}}_{\mathit{in}}(t)$, $t\in \left[0,5\right]\mathrm{s}$. Bottom left panel: blue curve, average cell normal surface velocity, $v_{\mathit{cell}\text{,}n}(t)$ ; red curve, percentage volume variation, $\mathrm{\Delta }{𝒱}_{\%}\left(t\right)$, $t\in \left[0,5\right]\mathrm{s}$. Bottom right panel: total AH volumetric flow rate $Q_{\mathrm{A}\mathrm{H}}(t)$ for $t\in \left[0,5\right]\mathrm{s}$.

Figure 12.

Top left panel: a zoom of the membrane potential ${\psi }_m(t)$ (units: mV) in the time interval $t\in \left[0,10\right]\mathrm{s}$. Top right panel: a zoom of $c_{{\mathrm{N}\mathrm{a}}^{+},\mathit{in}}(t)$ (blue curve), $c_{{\mathrm{K}}^{+},\mathit{in}}(t)$ (red curve), and $c_{{\mathrm{C}\mathrm{l}}^{-},\mathit{in}}(t)$ (green curve) (units: mM) in the time interval $t\in \left[0,120\right]\mathrm{s}$. Middle left panel: a zoom of the chlorine molar flux density $j_{{\mathrm{C}\mathrm{l}}^{-}}^{c\mathrm{*}}(t)$ (units: mM ms⁻¹) in the time interval $t\in \left[0,10\right]\mathrm{s}$. Middle right panel: a zoom of $v_{\mathit{cell},n}(t)$ (blue curve) and $\mathrm{\Delta }{𝒱}_{\%}(t)$ (red curve) in the time interval $t\in \left[0,120\right]\mathrm{s}$. Bottom center panel: a zoom of the total AH volumetric flow rate in the time interval $t\in \left[0,300\right]\mathrm{s}$.

Figure 13.

Left panel: total AH volumetric flow rate $Q_{\mathrm{A}\mathrm{H}}$ for $t\in \left[0,600\right]s$ as a function of ${\sigma }_X$. The black dashed line indicates the physiological value of $Q_{\mathrm{A}\mathrm{H}}$, equal to 2.75 μL min⁻¹, when $\mathrm{I}\mathrm{O}\mathrm{P}=15\mathrm{m}\mathrm{m}\mathrm{H}\mathrm{g}$. Right panel: the total osmo-oncotic pressure difference $\mathrm{\Delta }\mathrm{\Pi }(t)$ for $t\in \left[0,600\right]\mathrm{s}$ as a function of ${\sigma }_X$.

Figure 14.

Total AH volumetric flow rate $Q_{\mathrm{A}\mathrm{H}}$ for $t\in \left[0,5400\right]\mathrm{s}$ as a function of $M_{ATP}$. The black dashed line indicates the physiological value of $Q_{\mathrm{A}\mathrm{H}}$, equal to 2.75 μL min⁻¹, when $\mathrm{I}\mathrm{O}\mathrm{P}=15\mathrm{m}\mathrm{m}\mathrm{H}\mathrm{g}$.

Figure 15.

Plot of $\mathrm{\Delta }{Q}_{\%}(\mathrm{I}\mathrm{O}\mathrm{P})$ for IOP in interval [15, 150] mmHg, with $t\in \left[0,5400\right]\mathrm{s}$.